A few notes about proofs – Math 419, Fall 2009
A few students asked if I could help them with proofs from the homework
assignments as they prepared for the first exam.
The purpose of this document is to show you how to get rolling on a few of
the proofs from the homework.
1.2/2a: For these sorts of basic manipulation proofs, it’s usually OK just to
write out definitions and show that you can get from the left side of the equation
to the right side. Sometimes it helps to work the problem from both directions
– one direction may be easier than the other. For this one, just use z = x + iy
and do the computation.
1.2/4: To verify something like this, just plug it in and make sure it works.
1.3/4: This one feels a bit like induction in that you know something similar
and would probably benefit from using that other fact. As with induction, it’s
just a matter of making the problem look like the case you know.
1.5/10: Same reasoning as 1.2/2a.
1.8/9: On this one, you do a computation of some sort, but the proof you
are looking for comes from taking the real parts of each. It’s worth keeping
in mind that the real and imaginary parts of a function are both real-valued
functions of two real variables.
1.10/7: This one is really easy if you use the stuff you know (see the hint).
Don’t hesitate to use theorems that you know, as long as you state that that’s
what you are doing,
1.11/9,10: Good stuff to know, but don’t sweat it for the exam.
2.18/4: I like induction. Remember – you only need to show one base case.
In the induction step, make the thing that you are trying to prove look just like
the thing you are assuming is true....
2.18/5: It’s good to know how to show that a function is non-continuous
somewhere (by looking along two paths and seeing that the limits do not agree).
2.20/8: This is sort of like showing that a function is discontinuous, and
it’s a good technique to be aware of. That is, just choose two different paths in
(δx, δy) space that lead to two different limits.
2.23/1,5 and 2.25/1,2: The key to all of these is the Cauchy-Riemann
equations. To see where a function is analytic, you need to check where the
CREs hold (and where the partials are continuous).
2.25/7 and 2.26/2,3: Here again, the key is the Cauchy-Riemann equations. In each case, you are given a problem that has implications for the partials
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of an analytic function. You can use the Cauchy-Riemann equations and real
integration to recover the properties of the function stated in the problem.
2.28/1: This is just a straightforward application of the theorem of section
27. Remember that that theorem holds for both regions and line segments.
3.29/14: Don’t worry about part (b) sorts of arguments for the exam. (a)
is another nice version of induction.
3.34/2,5: These are fairly straightforward plug and chug problems, just in
the setting of a proof.
I think that’s it. The rest of the problems were fairly computational in
nature. Please review those problems, too; the point of this document is just to
summarize how each of the proofs on the homework went.
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